Answer
Does not exist
Work Step by Step
Given
$$ \lim _{(x, y) \rightarrow(0,0)} \frac{x^{2}-y^{2}}{x^{2}+y^{2}}$$
Let $ y=mx$, then
\begin{align*}
\lim _{(x, y) \rightarrow(0,0)} \frac{x^{2}-y^{2}}{x^{2}+y^{2}}&=\lim _{x\rightarrow0} \frac{x^{2}-m^{2}x^2}{x^{2}+m^{2}x^2}\\
&=\lim _{x\rightarrow0} \frac{1-m^{2} }{ 1+m^{2} }\\
\end{align*}
Since the limit depends on $m$, then $ \lim _{(x, y) \rightarrow(0,0)} \frac{x^{2}-y^{2}}{x^{2}+y^{2}}$ does not exist.
